Apr 13, 2009 - ticular, we elaborate how to achieve the wanted operations to create and manipulate the topological quantum states, providing an ...
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PHYSICAL REVIEW A 79, 040303共R兲 共2009兲
Implementing topological quantum manipulation with superconducting circuits 1
Zheng-Yuan Xue,1 Shi-Liang Zhu,1,2 J. Q. You,3 and Z. D. Wang1
Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 2 Laboratory of Quantum Information Technology, ICMP and SPTE, South China Normal University, Guangzhou 510006, China 3 Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China 共Received 16 January 2009; published 13 April 2009兲 A two-component fermion model with conventional two-body interactions was recently shown to have anyonic excitations. We here propose a scheme to physically implement this model by transforming each chain of two two-component fermions to the two capacitively coupled chains of superconducting devices. In particular, we elaborate how to achieve the wanted operations to create and manipulate the topological quantum states, providing an experimentally feasible scenario to access the topological memory and to build the anyonic interferometry. DOI: 10.1103/PhysRevA.79.040303
PACS number共s兲: 03.67.Lx, 42.50.Dv, 85.25.Cp
Topological ordered states emerge as a new kind of states of quantum matter beyond the description of conventional Landau’s theory 关1兴, whose excitations are anyons satisfying fractional statistics. A paradigmatic system for the existence of anyons is a kind of so-called fractional quantum Hall states 关2兴. Alternatively, artificial spin lattice models are also promising for observing these exotic excitations 关1,3,4兴. Kitaev models 关3,4兴 are most famous for demonstrating anyonic interferometry and braiding operations for topological quantum computation. Since anyons have not been directly observed experimentally, a focus at present is to experimentally demonstrate the topological nature of these states. Abelian anyons maybe relatively easy to achieve and to manipulate in comparison with the nonabelian ones, thus it is of current interest to explore them both theoretically and experimentally. Kitaev constructed an artificial spin model 关3兴, i.e., the toric code model, which supports the abelian anyon. But the wanted four-body interactions are notoriously hard to generate experimentally in a controllable fashion. Alternatively, it was proposed 关5兴 to generate dynamically the ground state and the excitations of the model Hamiltonian, instead of direct ground-state cooling, to simulate the anyonic interferometry. On the other hand, implementation of another Kitaev’s honeycomb model 关4兴 was also suggested in the context of ultracold atoms 关6兴, polar molecules 关7兴, and superconducting circuits 关8兴. The honeycomb model 关4兴 is an anisotropic spin model with three types of nearest-neighbor two-body interactions, which support both abelian and nonabelian anyons. It was shown 关4兴 that the toric code model can be obtained from the limiting case of the honeycomb model. Using this map, preliminary operations for topological quantum memory and computation were also addressed 关9–12兴. Nevertheless, in this case, anyons are created by the fourth-order perturbation treatment, which would blur the extracted anyonic information 关12兴. In addition, this map is good but not exact, so that one may get both anyonic and fermionic excitations. Therefore, new methods for implementing the model and manipulating the relevant topological states are still desirably awaited. Recently, a two-component fermion model 关13兴 with conventional two-body interactions was shown to have anyonic 1050-2947/2009/79共4兲/040303共4兲
excitations, which obey the same fusion rules as those of the toric code model and are mutual semions. This model is promising because it provides an example for abelian anyons, which can be directly implemented. In this Rapid Communication, we propose to physically implement this model with appropriately designed superconducting circuits. In particular, we elaborate how to achieve the wanted operations that create and manipulate the topological states as well as anyons with the cavity-assisted interactions using an external magnetic drive, providing an experimentally feasible scenario to access the topological memory and to build the anyonic interferometry. The Hamiltonian for the two-component fermion model in a two-dimensional square lattice, as shown in Fig. 1共a兲, is 关13兴 H f1 = − Jq 兺 共2n↑,i − 1兲共2n↑,j − 1兲 具i,j典
− J p 兺 共2n↓,i − 1兲共2n↓,j − 1兲 具i,j典
+ U 兺 共2n↑,i − 1兲共2n↓,i − 1兲,
共1兲
i
(a )
( b)
i
j
3 (c) i
iw ib
4
2 (d ) 1
iw ib
jw jb
5
6
FIG. 1. 共Color online兲 A map from a square lattice to a honeycomb lattice by extending each square lattice site to be a vertical link. 共a兲 The square lattice, where 具i , j典 denotes the nearest neighbors along the horizontal diagonal direction. 共b兲 The honeycomb lattice, where ib,w label the black and white sublattices. The indicated zigzag chain is one of the lines for the Jordan-Wigner transformation. 共c兲 Each square lattice site being extended as a vertical link of Majorana fermions with iw 共ib兲 labels the white 共black兲 sublattice and iw on the top of ib. 共d兲 Site labels within a plaquette.
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©2009 The American Physical Society
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XUE et al. † where ns,i = cs,i cs,i and cs,i are annihilation operators of spin-s fermions and 具i , j典 mean the nearest neighbors along the horizontal diagonals of squares. The ground states of this Hamiltonian are highly degenerated, i.e., every individual chain is ferromagnetic. As shown in Ref. 关13兴, the low-lying excitations are deconfined mutual semions under the open boundary condition. To show the nature of the low-lying excitations, we map the square lattice to the honeycomb lattice, as shown in Fig. 1共b兲, by extending each lattice site in the square lattice to be a vertical link of Majorana fermions 关see Fig. 1共c兲兴 defined † † † 兲, iw = c↑,i + c↑,i ; ib = −i共c↓,i − c↓,i 兲, and by ib = −i共c↑,i − c↑,i † iw = c↓,i + c↓,i. These “real” fermion operators obey i2 = i2 = 1. Otherwise, they are anticommutative. In this s s Majorana fermion representation, the Hamiltonian 共1兲 is mapped to
˜ + U兺 Q Q ˜ H f2 = − Jq 兺 Wij − J p 兺 W ij i i, 具ij典
具ij典
共2兲
i
˜ =Q ˜Q ˜ and W ij i j with
where Wi,j = QiQ j with Qi = iiwib ˜ = i . Since there is no coupling between the chains, Q i iw ib the present model may be transferred to a spin model ˜ − U 兺 Sz Sz , H f3 = − Jq 兺 W P − J p 兺 W P i i P
P
i
b
w
共3兲
by using Jordan-Wigner transformation 关14兴: jw z z z = Syj 兿 j⬘⬍jwS j⬘, jb = Sxj 兿 j⬘⬍jbS j⬘, jw = Sxj 兿 j⬘⬍jwS j⬘, and jb w w b s s s s s s z = Syj 兿 j⬘⬍jbS j⬘, where Sx,y,z are the corresponding Pauli matrib
s
s
ces for the defined Majorana fermions, W P = S1y Sx2Sz3S4y Sx5Sz6 ˜ = Sx Sy Sz Sx Sy Sz with the site labels within a plaquette and W P 1 2 3 4 5 6 depicted in Fig. 1共d兲. The order of the sites is defined as follows: js ⬎ jt if the zigzag line 关one of such lines is indicated in Fig. 1共b兲兴 including js is higher than that of jt, or if js is on the right hand of jt when they are in the same line. It ˜ 兴 = 0 for every plaquette, is straightforward to check 关W P , W P⬘ and all of them also commute with the third term. In this spin model, the ground state can be written as ˜ 兲兩典, 兩G典 = 兿 共1 + W P兲共1 + W P
Vg Cg
Cc
Ca Cb
b Vg
FIG. 2. 共Color online兲 A schematic circuit of two chains of capacitively coupled superconducting devices, labeled by a and b, to implement a chain of two-component fermions. Here only the first device of the two chains is explicitly shown, while others are simply denoted as the filled circles with different colors 共shades兲 for different chains.
small superconducting island with n excess Cooper pair charges connected by a Josephson junction with coupling energy EJ and capacitance CJ. A control gate voltage Vg is applied via a gate capacitor Cg. To quantize the circuit equation, we first introduce the Hamiltonian and then convert the classical momentum variable to the momentum operator. Then the Hamiltonian reads Hq1 = 兺
冋
册
␣Ct 2 共˙ 兲 − Ej cos − ␣C˙ a˙ b , 2
共5兲
where is the gauge phase drop across the corresponding junction, Ct = C0 + Cc with C0 = Cg + CJ, ␣ = 共ប / 2e兲2, and the induced charge ng = CgVg / 2e. At temperatures much lower than the single-pair charging energy, i.e., kBT Ⰶ Ec = e2 / 共2C0兲, and restricting the gate charge to the range of ng 苸 关0 , 1兴, only a pair of adjacent charge states 兵兩0典, 兩1典其 on the island are relevant. The Hamiltonian 共5兲 is then reduced to 关16兴 Hq2 = −
共4兲
P
where 兩典 = 兩1 ¯ 1典 is a reference state and each “1” means the eigenvalue of Szj being 1. Similar to the Kitaev’s honb共w兲 eycomb model 关4兴, all excitations here may be labelled by ˜ . The low-energy excitatwo quantum numbers W P and W P tions fall into two closed subsets, each can be graded by a Z2 ⫻ Z2 group. The fusion rules of these excitations are equivalent to the excitations in the toric code model 关3兴, and different graded vortices are mutual abelian semions 关15兴. We now proceed to implement the model with capacitively coupled superconducting devices, i.e., the Cooper pair box. The key idea is to use two chains of capacitively coupled superconducting devices, as shown in Fig. 2, to implement a chain of two-component fermions. A building block of our implementation, as shown in the rectangle of Fig. 2, is the two capacitively coupled superconducting devices. A typical design of a Cooper pair box consists of a
a
1 兺 关⑀共1 − 2ng兲z + ⌬x兴 + zazb , 2
共6兲
+ Cc兲 / ⌳ with ⌳ = Cat Cbt − C2c , ⌬ = EJ, where ⑀a共b兲 = 2e2共Cb共a兲 t 2 = e Cc / ⌳, 苸 兵a , b其 and x,z denotes the corresponding Pauli matrix in the basis of 兵兩0典, 兩1典其. The single-device terms in Hamiltonian 共6兲 can be tuned to be zero by conventional methods 关17兴. Therefore, in what follows, we do not take them into consideration. For two identical devices 共C0 = C0兲, ⑀ = 4Ec and = e2Cc / 共C2t − C2c 兲 ⯝ 2Ec with  = Cc / C0. It is notable that the strength of this interaction, proportional to the coupling capacitance, is stronger than any other presentknown coupling methods. The circuit Hamiltonian of the two coupled chains, as depicted in Fig. 2, can be obtained in a similar way. In this extended multipartite coupling case, the long-range interaction between the devices would appear, which decays exponentially as 兩i−j兩 with i and j being the site labels of the two involved devices 关18兴. Therefore, the long-range interaction
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IMPLEMENTING TOPOLOGICAL QUANTUM MANIPULATION…
y
o
x
FIG. 3. 共Color online兲 A schematic diagram of the cavityassisted manipulation. The x and y coordinates are denoted by the arrows, and the z direction is pointing out the xoy plane. The square lattice 共red兲 is placed to be parallel to the yoz plane. The square lattice and the auxiliary device 共blue rectangle兲 are placed with the x coordinate at the antinodes of the single-mode standing-wave cavity. All superconducting devices are placed with their loop plane being parallel to the xoz plane, which is perpendicular to the magnetic component of the cavity field, letting it be the only contributed component.
is negligible if  Ⰶ 1, and in typical experiments  ⯝ 0.05 关19兴. Up to the first order of , the interaction Hamiltonian is given by HJJ = 兺 z;jz;共j+1兲 + 兺 caz;jbz;j , ;j
共7兲
j
where ⯝ e2C / 关C0 + 2共Cc + 2C兲兴 and c ⯝ e 2C c / 关C0 + 2共Cc + Ca + Cb兲兴. Once the coupled chains are placed according to the geometry of Fig. 1共a兲, a corresponding two-dimensional square lattice model is constructed. Drawing an analogy between the device states in chain a 共b兲 and spin ↑ 共↓兲, the above interaction Hamiltonian in the addressed system may be rewritten as the two-component fermion model of Eq. 共1兲, with the parameters 共a , b , c兲 corresponding to 共Jq , J p , U兲 关20兴. As a result, the topologically protected ground state of Eq. 共4兲 may be implemented with the present setup of superconducting devices. In addition, to accomplish certain topological quantum manipulation tasks, including the examination of the anyon statistics, it is a must to have a set of basic operations of the devices. In the Majorana fermion representation, the wanted operations are 关13兴 Szj = i jb jb, b
Sxj = jb 兿 共i j j兲, b
j⬍jb
Szj = i jw jw ,
共8a兲
Sxj = jw 兿 共i j j兲,
共8b兲
w
w
manipulate the states, we put the lattice into a microwave cavity, with the geometry of the hybrid system being explained in the caption of Fig. 3. For simplicity, we consider only the single-mode standing wave cavity. To be more specific, we reach the cavity-assisted manipulation by a magnetic drive 关21兴. The interaction can be switched on/off by modulating the external magnetic field to be ac/dc 关21兴. With the dc magnetic flux, the external flux is merely used to address separately the single-qubit rotations. Under the cavity field, it is also readily possible to tune off single-qubit terms. The wanted operations in Eq. 共8a兲 for selected device can be achieved with the cavity-mediated integrations by tuning the driven magnetic flux of the device to be of ac. The Pauli matrices Szjw and Szjb in Eq. 共8a兲 correspond to x 丢 x and y 丢 y two-body interaction of two devices of jth site, respectively. These two type interactions for each lattice site can be directly engineered in our hybrid implementation 关21兴 as it allows selected addressing of designated devices. These two interactions are mediated by the virtue cavity photon, thus we need to keep the cavity mode in the vacuum state. Here, the devices work in their degeneracy points. The common cavity mode can also be used to realize the global stringlike Sx operators in Eq. 共8b兲. The off-resonant interaction between the cavity mode and the selected devices is 关22兴 HQND = nc 兺 zj ,
where nc = a†a is the photon number operator of the cavity mode, the coupling strength is = g2 / 2␦ with g as the singlephoton Rabi frequency for the cavity mode, and ␦ is the detuning between the cavity mode frequency c and optical transition frequency in atomic spins. In our implementation, this can be the 兩1典 → 兩2典 transition of the selected devices, where 兩2典 is an ancillary energy level beyond the qubit subspace 兵兩0典, 兩1典其, and the frequency of the drive ac flux satis˜ 12 + c + ␦ with ˜ 12 = 2Ec共3 − 2ng兲 / ប. To avoid the fies = transitions 兩0典 → 兩1典 and 兩0典 → 兩2典 by the ac drive, we ˜ 01 = 2Ec共1 − 2ng兲 / ប via ng so that ␦ Ⰶ ⌬1,2, where tune ˜ 01 and ⌬2 = ˜ 01 + ˜ 12 − are the corresponding de⌬1 = − tunings. This quantum nondemolition 共QND兲 Hamiltonian 共9兲 preserves the photon number of the cavity mode. Within the nc 苸 兵0 , 1其 subspace, the evolution of the QND Hamiltonian during the interaction time = / 2 yields 关10兴
j⬍jw
where Szj is the spin-flip operator for a given site and it also transfers a double fermion occupancy to empty or vice versa. Sxj denotes a nonlocal operation that creates or annihilates a fermion at site j and changes the site occupation for sites j ⬍ jb共w兲 关13兴. In the present model, Sz and Sx are effective Pauli matrices, which, according to Ref. 关4兴, may create and move the excitations. At this stage, let us elaborate how to obtain the wanted operations in Eqs. 共8a兲 and 共8b兲. Notably, individual addressability is normally a prerequisite in such manipulation. In the present proposal, the size of the device setup is macroscopic, thus individual addressability is taken as granted. To
共9兲
j
U = exp关− iH兴 =
再
I 共− i兲
N
兿j
for nc = 0
zj
for nc = 1
冎
,
共10兲
where N is the number of the selected devices. From Eq. 共10兲, 共controlled兲 string operations for an arbitrary string can be achieved 关10兴. If the cavity is initially prepared in the nc = 1 state, the global operation reduces to the string operation Uz = 兿 jzj . As all string operators are equivalent to Uz up to local single spin rotations 关10兴, all string operations for arbitrary string can be achieved: Ux = 兿 jxj = HUzH and Uy = 兿 jyj = RUzR, where H = 兿 jH j and R = 兿 jR j with H j = 共xj + zj 兲 / 冑2 being the Hadamard rotation and R j = exp共−i 4 zj 兲. Therefore, with this
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elementary operation, creation and manipulation of anyons are likely feasible in our scheme. For example, Sxj = jb b
兿
kb⬍jb
= by;j
兿 k ⬍j b
共ikbkb兲
兿
kw⬍jb
a b 共ix;k x;k 兲 b
b
b
共ikwkw兲
兿 k ⬍j w
共iay;k by;k 兲 w
w
共11兲
b
denotes a nonlocal operation that could create a domainwall-like excitation/semion 共at the site j兲 from the ground states under the open boundary condition 关13兴. If the cavity is initially prepared in a superposition of zero- and one-photon states, the global operation in Eq. 共10兲 reduces to a controlled-string operation: Ucs = 兩0典具0兩 丢 I + 兩1典具1兩 丢 Uz, where the parameters and are controlled by the initially prepared photon number state. With such a controlled-string operation, one is able to access the topological memory and to build anyonic interferometry 关10兴. In simulating the string operations, we need to engineer the cavity number states. Therefore, beside the square lattice, we also place an ancilla device in the cavity, as shown in Fig. 3, which is used to control the cavity photon number state by swapping its states with that of the cavity using the resonate
关1兴 X.-G. Wen, Quantum Field Theory of Many-Body Systems 共Oxford University Press, New York, 2004兲. 关2兴 C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 共2008兲. 关3兴 A. Kitaev, Ann. Phys. 共N.Y.兲 303, 2 共2003兲. 关4兴 A. Kitaev, Ann. Phys. 共N.Y.兲 321, 2 共2006兲. 关5兴 Y.-J. Han, R. Raussendorf, and L.-M. Duan, Phys. Rev. Lett. 98, 150404 共2007兲. 关6兴 L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 共2003兲. 关7兴 A. Micheli, G. K. Brennen, and P. Zoller, Nat. Phys. 2, 341 共2006兲. 关8兴 J. Q. You, X.-F. Shi, and F. Nori, e-print arXiv:0809.005. 关9兴 C. Zhang, V. W. Scarola, S. Tewari, and S. Das Sarma, Proc. Natl. Acad. Sci. U.S.A. 104, 18415 共2007兲. 关10兴 L. Jiang, G. K. Brennen, A. V. Gorshkov, K. Hammerer, M. Hafezi, E. Demler, M. D. Lukin, and P. Zoller, Nat. Phys. 4, 482 共2008兲. 关11兴 M. Aguado, G. K. Brennen, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 101, 260501 共2008兲. 关12兴 S. Dusuel, K. P. Schmidt, and J. Vidal, Phys. Rev. Lett. 100, 177204 共2008兲. 关13兴 Y. Yu and Y. Li, e-print arXiv:0806.1688. 关14兴 X.-Y. Feng, G.-M. Zhang, and T. Xiang, Phys. Rev. Lett. 98, 087204 共2007兲; H.-D. Chen and Z. Nussinov, J. Phys. A: Math. Theor. 41, 075001 共2008兲.
cavity-device interaction. This swap operation can be achieved by the famous Jaynes-Cummings model Hamiltonian: HJC = ⍀共a+ + a†−兲, which can be implemented in our system by choosing the frequency of the ac driven mag˜ 01. In netic flux for the ancillary device satisfying = c + this case, we need to tune the device slightly away from the degeneracy point, which results in a shorter decohenrence time. Fortunately, the resonate operation is also much faster. In summary, we have proposed an exotic scheme to implement a two-component fermion model using superconducting quantum circuits, which was shown to support abelian anyonic excitations. Most intriguingly, we have elaborated how to achieve all the wanted operations that could create and manipulate the anyonic states. Our approach provides an experimentally feasible scenario to access the topological memory and to build the anyonic interferometry. We thank Yue Yu, L. B. Shao and Liang Jiang for many helpful discussions. This work was supported by the RGC of Hong Kong under Grants No. HKU7045/05P and No. HKU7049/07P, the URC fund of HKU, and NSFC under Grants No. 10429401, No. 10674049, and No. 10625416, and the National Basic Research Program of China under Grants No. 2006CB921800, No. 2007CB925204, and No. 2009CB929300.
关15兴 Although it is hard to figure out the presence of anyons directly from Hamiltonian 共1兲, it is quite clear from the theory of Kitaev 关4兴 that Hamiltonian 共3兲 does support anyonic excitations from the ground state denoted by Eq. 共4兲. 关16兴 J. Q. You, X. Hu, and F. Nori, Phys. Rev. B 72, 144529 共2005兲. 关17兴 J. Q. You and F. Nori, Phys. Today 58共11兲, 42 共2005兲; Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 共2001兲. 关18兴 G. P. Berman, A. R. Bishop, D. I. Kamenev, and A. Trombettoni, Phys. Rev. B 71, 014523 共2005兲. 关19兴 Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Phys. Rev. Lett. 87, 246601 共2001兲; Yu. A. Pashkin, T. Yamamoto, O. Astaffev, Y. Nakamura, D. V. Averin, and J. S. Tsai, Nature 共London兲 421, 823 共2003兲. 关20兴 It is noted that the signs before a and b in Hamiltonian 共7兲 are intially positive. However, the signs can be tuned to be minus by a proper redefinition of the analogy between device states 兵兩0典, 兩1典其 and pseudospin states 兵兩⫹典, 兩⫺典其, i.e., 兩0典 共兩1典兲 corresponds to 兩⫹典 共兩⫺典兲 for all odd-number sites while 兩0典 共兩1典兲 corresponds to 兩⫺典 共兩⫹典兲 for all even-number sites. 关21兴 S.-L. Zhu, Z. D. Wang, and P. Zanardi, Phys. Rev. Lett. 94, 100502 共2005兲; Z.-Y. Xue, Z. D. Wang, and S.-L. Zhu, Phys. Rev. A 77, 024301 共2008兲; e-print arXiv:0806.0753. 关22兴 M. O. Scully and M. S. Zubairy, Quantum optics 共Cambridge University Press, New York, 1997兲.
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